#### ∫tan x tan 2x tan 3x dx is equal to ____.

tan 3x = tan (x + 2x) = tanx + tan2x/1-tanxtan2x On cross multiplying, tan 3x - tan3x tan x tan 2x = tan x + tan 2x tan 3x tan 2x tan x = tan 3x - tan x - tan 2x ∫tan3xtan2xtanxdx=∫tan3xdx-∫tanxdx-∫tan2xdx = 1/3 log|sec 3x|- log|sec x| - 1/2 log |sec 2x| + c

#### ?sinax + bdxcan be represented as ____.

(d/dx) (?cos(ax + b)+ c)dx = - a?sin(ax + b)dx In this, differentiation and integration annual each other, and we get cos (ax + b) + c = - a?sin(ax + b)

#### In indefinite integration, ∫f(x)dx = g(x)+ c, c is called the constant of integration and it can take ____.

Integration of the function is over an interval. Hence, c can take infinitely many values but not any value, since c depends on various factors.

#### The primitive of f(x) g"(x) - f"(x) g(x) is equal to ____.

∫(f(x) g"(x) - f"(x) g(x)) dx = ∫f(x) g"(x) dx - ∫f"(x) g(x) dx = f(x) g'(x) - ∫f'(x) g'(x) dx - g(x) f'(x) + ∫f'(x) g'(x) dx = f(x) g'(x) - g(x) f'(x) + c

#### Integration and differentiation are inverse of each other. This statement can be represented mathematically as ____.

The statement 'Integration and differentiation are inverse of each other’ means if they occur together, they neutralize each other. This is reflected in the option A.

#### If d/dx f (x) = F (x), then the expression can be written in the integral form as ____.

Given the derivative of the function, the original function or the primitive of the function can be determined using integration. Hence, by definition? F(x)dx = f(x).

#### The derivative of a function can be determined at a point, bu integration can be done ____.

By definition, integration is the summation of infinitely large number of infinitesimally small quantities. Therefore, integration can be done over an interval.

#### If y = mx^{2} + c represents a family of curves, then the family can be written using notation of integration as ____.

The indefinite integral of a function represents geometrically, a family of curves having parallel tangents at their points of intersection with the lines perpendicular to the axis representing the variable of integration. The variable of integration in this integral is x. ∫2mxdx = 2m x^{2}/2 + c = mx^{2}+c

#### If the derivative of e^{x} cos x is e^{x} (cos x - sin x), then this can be written using the integral symbol as ____.

By definition, e^{x} cos x is the anti derivative of ∫ e^{x} (cos x - sin x) dx. (d/dx) e^{x} cosx = e^{x} (cosx-sinx)

#### The differential equation dy/dx = kx represents a family of curves. If the line x = a intersects each of the curve of the family of curves at distinct points, and if tangents to each of the curve is drawn at these points of intersection, then these tangents will ____.

dy/dx = kx:dy = kxdx ∫dy = ∫kx dx y = k x^{2}/2 + c By varying c, we get the family of curves. And, the slope of tangent to the curve is given by the differential equation dy/dx = kx And, slope of the tangents at the point of intersection of these curves is given with the line x = a = ka. This indicates that the tangents are parallel at these points.

#### Let ∫f(x)dx = g(x). If G(x) be any other integral of f(x), then G(x) = ____.

∫f(x) dx = g(x) and ∫f(x) dx = G(x) By definition, g'(x) = f(x) and G'(x) = f(x) Define h(x) = G(x) - g(x) h'(x) = G'(x) - g'(x) = f(x) - f(x) = 0 Let a and b be any 2 real numbers such that h(x) continuous on [a, b], derivable on (a, b). Then, by legranges mean value theorem, there exists at least one c ∈ (a, b) such that h(b)-h(a)/b-a = h'(c) h(b) - h(a) = b - a(h’(c)) = b - a(0) = 0 h(a) = h(b) Then, h(x) should be a constant function. G(x) - g(x) is a constant function. G(x) - g(x) = c or G(x) = g(x) + c

#### If f and g are functions of x such that g’(x) = f(x), then the function g is called ____.

d/dx g(x) = f(x) i.e. ∫f(x) dx = g(x) + c, where c is called the constant of integration (Because Derivative of (g(x) + c)' = g'(x)) If f and g are functions of x such that g’(x) = f(x), then the function g is called anti derivative of f with respect to x.

#### ∫ e^{x} sin x dx is equal to ____.

∫ e^{x} sin x dx = e^{x} sin x - ∫ e^{x} cos x dx Applying parts rule again, we get ∫ e^{x} sin x dx = e^{x} sin x – e^{x} cos x + ∫- e^{x} sin xdx 2∫ e^{x} sin x dx = e^{x} (sin x - cos x) + c ∫ e^{x} sin xdx = e^{x} /2 (sin x-cos x)+c

#### ∫(log(logx.)/x) dx is equal to ____.

Let log x = t. Then, 1/x dx = dt ∫(log(logx)/x)dx=∫logtdt =∫logt(1)dt=tlogt-∫t/tdt = t log t - t + c = log x log(log x) - log x + c